In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

DeepSDF: Learning Continuous Signed Distance Functionsfor Shape Representation.

Let ψ: X Y be an arbitrary G equivariant function. We leave proving this as a future work. We now show the following: Proposition 3. The proposed distribution p We now show the following: Proposition 6. From Eq. (29), we have: ϕ Proposition 7. The proposed symmetrization From Eq. (29), we have: ϕ This is after handling the translation component of the Euclidean group E ( d) / SE (d) as in Eq. (29). We now show the following: Proposition 8. Therefore, probabilistic symmetrization can become frame averaging.

Canonicalization is a widely used strategy in equivariant machine learning, enforcing symmetry in neural networks by mapping each input to a standard form. Yet, it often introduces discontinuities that can affect stability during training, limit generalization, and complicate universal approximation theorems. In this paper, we address this by introducing adaptive canonicalization, a general framework in which the canonicalization depends both on the input and the network. Specifically, we present the adaptive canonicalization based on prior maximization, where the standard form of the input is chosen to maximize the predictive confidence of the network. We prove that this construction yields continuous and symmetry-respecting models that admit universal approximation properties. We propose two applications of our setting: (i) resolving eigenbasis ambiguities in spectral graph neural networks, and (ii) handling rotational symmetries in point clouds. We empirically validate our methods on molecular and protein classification, as well as point cloud classification tasks. Our adaptive canonicalization outperforms the three other common solutions to equivariant machine learning: data augmentation, standard canonicalization, and equivariant architectures.

Symmetry-aware methods for machine learning, such as data augmentation and equivariant architectures, encourage correct model behavior on all transformations (e.g. rotations or permutations) of the original dataset. These methods can improve generalization and sample efficiency, under the assumption that the transformed datapoints are highly probable, or "important", under the test distribution. In this work, we develop a method for critically evaluating this assumption. In particular, we propose a metric to quantify the amount of anisotropy, or symmetry-breaking, in a dataset, via a two-sample neural classifier test that distinguishes between the original dataset and its randomly augmented equivalent. We validate our metric on synthetic datasets, and then use it to uncover surprisingly high degrees of alignment in several benchmark point cloud datasets. We show theoretically that distributional symmetry-breaking can actually prevent invariant methods from performing optimally even when the underlying labels are truly invariant, as we show for invariant ridge regression in the infinite feature limit. Empirically, we find that the implication for symmetry-aware methods is dataset-dependent: equivariant methods still impart benefits on some anisotropic datasets, but not others. Overall, these findings suggest that understanding equivariance -- both when it works, and why -- may require rethinking symmetry biases in the data.

General-purpose robotic skills from end-to-end demonstrations often leads to task-specific policies that fail to generalize beyond the training distribution. Therefore, we introduce FunCanon, a framework that converts long-horizon manipulation tasks into sequences of action chunks, each defined by an actor, verb, and object. These chunks focus policy learning on the actions themselves, rather than isolated tasks, enabling compositionality and reuse. To make policies pose-aware and category-general, we perform functional object canonicalization for functional alignment and automatic manipulation trajectory transfer, mapping objects into shared functional frames using affordance cues from large vision language models. An object centric and action centric diffusion policy FuncDiffuser trained on this aligned data naturally respects object affordances and poses, simplifying learning and improving generalization ability. Experiments on simulated and real-world benchmarks demonstrate category-level generalization, cross-task behavior reuse, and robust sim2real deployment, showing that functional canonicalization provides a strong inductive bias for scalable imitation learning in complex manipulation domains. Details of the demo and supplemental material are available on our project website https://sites.google.com/view/funcanon.

Real world data often exhibits unknown, instance-specific symmetries that rarely exactly match a transformation group $G$ fixed a priori. Class-pose decompositions aim to create disentangled representations by factoring inputs into invariant features and a pose $g\in G$ defined relative to a training-dependent, arbitrary canonical representation. We introduce RECON, a class-pose agnostic $\textit{canonical orientation normalization}$ that corrects arbitrary canonicals via a simple right-multiplication, yielding $\textit{natural}$, data-aligned canonicalizations. This enables (i) unsupervised discovery of instance-specific symmetry distributions, (ii) detection of out-of-distribution poses, and (iii) test-time canonicalization, granting group invariance to pre-trained models without retraining and irrespective of model architecture, improving downstream performance. We demonstrate results on 2D image benchmarks and --for the first time-- extend symmetry discovery to 3D groups.